gives the spherical Bessel function of the first kind .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • SphericalBesselJ is given in terms of ordinary Bessel functions by .
  • SphericalBesselJ[n,z] has a branch cut discontinuity for noninteger in the complex plane running from to .
  • Explicit symbolic forms for integer n can be obtained using FunctionExpand.
  • For certain special arguments, SphericalBesselJ automatically evaluates to exact values.
  • SphericalBesselJ can be evaluated to arbitrary numerical precision.
  • SphericalBesselJ automatically threads over lists.


open allclose all

Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (37)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Limiting value at infinity:

SphericalBesselJ for symbolic n:

Find the first positive zero of SphericalBesselJ:

Different SphericalBesselJ types give different symbolic forms:

Visualization  (3)

Plot the SphericalBesselJ function for integer () and half-integer () orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

Function Properties  (12)

TemplateBox[{0, x}, SphericalBesselJ] is defined for all real and complex values:

TemplateBox[{{-, {1, /, 2}}, x}, SphericalBesselJ] is defined for all real values greater than 0:

Complex domain is the whole plane except :

Approximate function range of TemplateBox[{0, x}, SphericalBesselJ]:

For integer , TemplateBox[{n, z}, SphericalBesselJ] is an even or odd function in depending on whether is even or odd:

This can be expressed as TemplateBox[{n, z}, BesselJ]=(-1)^n TemplateBox[{n, {-, z}}, BesselJ]:

SphericalBesselJ threads elementwise over lists:

TemplateBox[{n, x}, SphericalBesselJ] is not an analytic function of for noninteger and negative values of :

SphericalBesselJ is neither non-decreasing nor non-increasing:

SphericalBesselJ is not injective:

SphericalBesselJ is neither non-negative nor non-positive:

TemplateBox[{n, z}, SphericalBesselJ] is singular for , possibly including , when is noninteger:

SphericalBesselJ is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z

Plot the higher derivatives with respect to z:

Formula for the ^(th) derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:


Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Use FullSimplify to simplify spherical Bessel functions of the first kind:

Recurrence relations:

Applications  (1)

Solve the radial part of the Laplace operator in 3D:

Properties & Relations  (2)

SphericalBesselJ can be represented as a DifferentialRoot:

Wolfram Research (2007), SphericalBesselJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalBesselJ.html.


Wolfram Research (2007), SphericalBesselJ, Wolfram Language function, https://reference.wolfram.com/language/ref/SphericalBesselJ.html.


Wolfram Language. 2007. "SphericalBesselJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalBesselJ.html.


Wolfram Language. (2007). SphericalBesselJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SphericalBesselJ.html


@misc{reference.wolfram_2021_sphericalbesselj, author="Wolfram Research", title="{SphericalBesselJ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalBesselJ.html}", note=[Accessed: 17-May-2022 ]}


@online{reference.wolfram_2021_sphericalbesselj, organization={Wolfram Research}, title={SphericalBesselJ}, year={2007}, url={https://reference.wolfram.com/language/ref/SphericalBesselJ.html}, note=[Accessed: 17-May-2022 ]}